There’s a difference between nicknames and diminutives. Diminutives are variations on a name that are based upon the sounds in that name. Bill, Billy, Will, etc. are diminutives for William. Calling the same person “Red” because his hair is read is using a nickname. Calling him “hotheaded Bill” is technically both
Using the name Kate, Kat, or Kathy in reference to someone named Katherine is also using a diminutive. Referring to my low-German ancestor Trientje by the name Katherine is not using a nickname or a diminutive. In this case it is a anglicization (translation into English) of the name Trientje. Your non-English speaking ancestors “new names” may not have been nicknames or diminutives at all–they could have been translations of their actual name.
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Many of my ancestors went by one of their middle names. Regina Katherine was known as Kate… and appears in census reports as such. Thank goodness for her Catherina Kunagunda got to be called Kate also!